Defender Van Dijk and the Reds defence kept another clean sheet in Saturday's 1-0 win at Crystal Palace to keep them top of the table heading into the international break.
The Dutch international is keen for Arne Slot's side to stay there and told Liverpool's website: 'We want to improve, we want to challenge, we want to compete until the final day and win and be successful.
'That's what we're focusing on - not what other expectations are. We've got a lot of points until now but we want to have a lot more.
'We know how tough the fixture list will be after the international break but why not enjoy it? I'm really looking forward to that and hopefully everyone involved with Liverpool is as well.'
Liverpool captain Virgil van Dijk is eager for his side to last the distance in the title race
Liverpool's defence has been key to their success this year, conceding just two Premier League goals so far.
However, goalkeeper Alisson suffered a hamstring injury and is expected to miss the Anfield clash with Chelsea on October 20.
Speaking about the Reds' rearguard, Van Dijk added: 'Obviously we keep going, we keep working and we're never satisfied. We always can improve.
'I think today we needed Ali twice, three times maybe, and that's what we're trying to avoid. But if we have to defend, we have to defend with everything that we've got.
'Everyone is contributing to clean sheets but I'm very happy with a clean sheet as well.'
Diogo Jota scored the only goal against Palace, who extended their winless run to seven matches.
The Eagles were unlucky not to be awarded a penalty after a tug by Van Dijk on Marc Guehi in the box.
Manager Oliver Glasner said: 'Why didn't VAR give it? They thought it wasn't a penalty. This situation isn't a discussion for me. It's not like handball or not. Guehi wants to go for the ball and is held by two hands, it influences his run, pulls him down.